Fundamental Theorem of Arithmetic. If a is an integer larger than 1, then
a can be written as a product of primes. Furthermore, this factorization is unique
except for the order of the factors.
proof: There are two things to be proved. Both parts of the p

Number Theory
The Greatest Common Divisor (GCD)
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(GCD)
1/1
Division Algorithm
Given integers a and b, with b > 0, there exists unique integers q (quotient)
and r (remainder) satisfying a = qb + r with 0 r < b.1
* the

Number Theory
Modular Arithmatic
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(Modular Arithmatic)
1/9
Congruence: definition
Let n be a positive integer. If a and b are integers, we say that
a is congruent to b modulo n,
denoted with a b (mod n),
if n|(a b) i

Proofs
Based on the principle of infinite descent
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(Proofs based on the principle of infinite descent)
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Principle
From the well-ordering principle, every nonempty set S of nonnegative
integers has a least element.

Proofs
Introduction
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(Intro to Proofs)
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Objective
what is a proof?
motivation for proofs
correctness of proof methods
role of elementary logic
examples for proof methods
(Intro to Proofs)
2 / 31
Outline
1
E

Sets, Relations, Functions
Equivalence relations to classify elements of a set
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(Classifying elements of a set)
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Definitions
If R is reflexive, symmetric, and transitive over set S then R is said to be
an equival

Number Theory
Linear Congruences
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(Linear Congruences)
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Linear congruence: definition
A congruence of the form ax b (mod m), where x is an unknown
integer, is called a linear congruence in one variable.
equivalen

Proofs
Well-ordered induction
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(Well-ordered induction)
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Well-ordering property
Every nonempty set of nonnegative integers has a least element.
based on the well-ordered set (Z + , )
(Well-ordered induction)
2/7
B

Sets, Relations, Functions
Element relation based set classification
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(Element relation based set classification)
1/7
Quasi-ordered
A set S with reflexive and transitive relation R over it said to be quasi-ordered
w.

Number Theory
Prime Numbers
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(Prime Numbers)
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Outline
1
Introduction
2
Infinitude of primes
(Prime Numbers)
2 / 14
Prime Number: definition
An integer p > 1 is called a prime number, or simply a prime, if its o

Sets, Relations, Functions
Cardinality based classification of sets
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(Cardinality based classification of sets)
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Outline
1
Finite vs infinite sets
2
Countably infinite vs uncountable sets
3
A hierarchy of uncou

Number Theory
Primality Testing
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(Primality Testing)
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Outline
1
Fermats little
2
Wilsons
(Primality Testing)
2/9
Fermats little theorem 1
If p is a prime and a is a positive integer with p 6 |a, then
ap1 1 (mod p